To be honest my knowledge about astronomy and related physics is close to zero. However, since I studied math a bit, I would like to ask a couple maybe stupid questions, which I am curious about and I don't know who and where to ask.

A) Projection of life

Premises:

As Carl Sagan explained in his "flatlanders" story, we can imagine high-dimension space by making projections into space with lower dimension.
As we learned at elementary school time is the fourth dimension.

Question:

Is it right that what we see/feel in every moment is projection of some 4D reality into our 3D world noticed by our senses.
For example, If my eyes were able to "somehow" see in 4D and I pointed my eyes to a person, would I see that person in all stages of his/her life from birth to death as a single "entity"? In every second we are only a projection of our 4D representation to our 3D world?

B) Parallel realities

Premises:

If we imagine 1D space as a line without ends, 2D space as an unbounded plane and 3D space as an unbounded space, then I could put infinity number of parallel lines into 2D space and infinity number of parallel planes to the 3D space.

I have read that there are at least 10 dimensions.

Questions:

Can I put infinite number of 3D spaces into a single 4D space? If yes, then if we for example chose geographic coordinates of Earth-Italy-Caesena as place and 49 B.C. as time coordinate, we would be probably close to event when Caesar screamed his famous: "alea iacta est".

If we considered that 5th dimension exists and that we could put infinite number of 4D spaces into 5D space, then we would end up with infinite number of parallel realities, in which Caesar did not crossed Rubicon, or did not even exist. Is it right or complete nonsense? Can be somehow proved that this is right/nonsense?

If we considered that 5th dimension exists and that we could put
infinite number of 4D spaces into 5D space, then we would end up with
infinite number of parallel realities, in which Caesar did not crossed
Rubicon, or did not even exist. Is it right or complete nonsense? Can
be somehow proved that this is right/nonsense?

Thinking of a 5D space-time as a collection of all possible 4D space-times is less suitable, since it would be a mere set of 4D space-times without meaningful structure. Take Ceasar crossing Rubikon a day earlier or day later or a meter to the left or a meter to the right with or without a leaf falling from a tree, or a huge number of other versions. Which would be the correct order to pile up these versions to a 5D world? The infinite-dimensional Hilbert space model solves this ambiguity, and provides a meaningful metric for the set of all possible worlds.

The 10-dimensional models of some string theories are something else: The 6 extra-dimensions are thought to be curled up, and tiny.

We have no access to parallel worlds, so it cannot be proven by observational evidence. It's a mathematical model. It's possible to prove theorems within the mathematical theory, but that's usually based on axioms, or assumptions. Mathematical theories may, but don't need to match with observation, depending on how the axioms match to physical "reality".

$\begingroup$'Projection' is a good term here to start with. In STR, time dilation and Lorentz contraction across different inertial frames can be described as due to projections to different temporal and spatial coordinate axes. The real difficulty is that this describes what we observe, which in STR is a very different concept from what we see or otherwise experience through our senses.$\endgroup$
– Stan LiouApr 24 '14 at 4:52

1

$\begingroup$@StanLiou I'd call this a 'change of base'. A projection would be kind of integrating along one or more dimensions, and get kind of lower-dimensional shadow. But no matter, whether we 'see' or 'observe' the universe at one instant, we consider just a lower-dimensional sheet or 'slice' of the 4D-space. That's why I feel uncomfortable with the use of 'projection' in this case. Of course, one could project a slice onto a different manifold of the same dimension, in order to get a description of the slice which may be more suitable for some purpose.$\endgroup$
– GeraldApr 24 '14 at 13:18

$\begingroup$You're correct in that the Lorentz transformation would be a change of basis, but for orthonormal bases appropriate to an inertial frame, the change of basis matrix itself consists of projection components $\langle e'_i,e_j\rangle$. Moreover, anytime one expresses a vector in an orthonormal basis, one take a sum of projections: $u = \sum_k e_k\langle e_k,u\rangle$, so different inertial frames disagreeing on the time component of a vector is exactly the statement that the projections to their respective temporal axes are different, and so forth for length contraction.$\endgroup$
– Stan LiouApr 24 '14 at 15:46

$\begingroup$@StanLiou I agree thus far. But we've still to slice the 4D-spacetime to get the observed spatial 3D-manifold (including contents) for a fixed time in the respective inertial frame.$\endgroup$
– GeraldApr 24 '14 at 21:08

$\begingroup$I suppose I'm confused about your unwillingness to use the term "projection" because you seem to be using it in a sense I don't recognize. I don't disagree with what you just said, but in what you describe, the mapping from spacetime to either the time or the spatial submanifold would both be projections.$\endgroup$
– Stan LiouApr 24 '14 at 23:14

Is it right that what we see/feel in every moment is projection of some 4D reality into our 3D world noticed by our senses.

Your and our world is not 3D, it is 4D, you just said it but then mysteriously you changed your approach. Our 'reality', our 'world' is 4D, there are 3 spacial dimensions and 1 time dimension. But you can live only in one point of space-time at a time.
Furthermore what you can see or experience at any given time is a slice of the space-time that we live in, the slice depends on your speed and point in space-time.

If we considered that 5th dimension exists and that we could put
infinite number of 4D spaces into 5D space, then we would end up with
infinite number of parallel realities

I think part 2 of your question has been answered. My short answer is yes! That's what string theory predicts.